Stability of compact actions and a result on divided differences

TL;DR

This paper investigates the stability of compact actions of al R^n on (n+1)-dimensional manifolds, providing conditions for perturbations that change orbit compactness, and introduces an estimate on divided differences.

Contribution

It generalizes previous stability results for smooth actions, offering new criteria for perturbations that alter orbit compactness and establishing an auxiliary estimate on divided differences.

Findings

Necessary and sufficient conditions for perturbations to change orbit compactness.

Existence of localized perturbations that alter orbit properties.

Examples of compact actions with perturbations affecting orbit compactness depending on smoothness class.

Abstract

We study smooth locally free actions of ${\mathbb R}^n$ on manifolds $M$ of dimension $n+1$. We are interested in compact orbits and in compact actions: actions with all orbits compact. Given a compact orbit in a neighborhood of compact orbits, we give necessary and sufficient conditions for the existence of a $C^k$ perturbation with noncompact orbits in the given neighborhood. We prove that if such a perturbation exists it can be assumed to differ from the original action only in a smaller neighborhood of the initial orbit. As an application, for each $k$, we give examples of compact actions which admit $C^{k-1}$-perturbations with noncompact orbits but such that all $C^k$-perturbations are compact. The main result generalizes for $k > 1$ a previous result for the case $C^1$. A critical auxiliary result is an estimate on divided differences.